Classical spinwave theory

Starting simple...

Spin waves are propagating disturbances of an ordered magnetic lattice.

Definition of a spin wave
Single ion magnetism

A classic spin wave

  • Spin is a vector of a fixed length
  • Equation of motion (Larmor precession):

$$ \frac{d\mathbf{S}}{dt} = \mathbf{S} \times \mathbf{h} $$
  • $\mathbf{h}$ field can be exchange (Weiss)field, external field or an anisotropy.
  • The general solution:

$$ \begin{align} dS_i^x & = A_i^x \cos (\omega t + \mathbf{k} \cdot \mathbf{r_i} + \varphi_i^x ) \\ dS_i^y & = A_i^y \cos (\omega t + \mathbf{k} \cdot \mathbf{r_i} + \varphi_i^y ) \end{align}$$
Classical spin waves in a ferromagnetic chain

Energy:


$$ E = \sum_i J \mathbf{S}_i \cdot \mathbf{S}_{i+1} $$

Equation of motion:


$$ \mathbf{S}_i = J \mathbf{S}_i \times ( \mathbf{S_{i-1}} + \mathbf{S_{i+1}} )$$

Assuming a small deviation from Equilibrium:


$$ \begin{align} \delta S_i^x & = JS (-\delta S_{i-1}^y -\delta S_{i+1}^y + 2\delta S_i^y) \\ \delta S_i^y & = JS (\delta S_{i-1}^y + \delta S_{i+1}^y - 2\delta S_i^y) \end{align}$$
Classical spin waves in a ferromagnetic chain
FM Spinwave dispersion
FM Spin chain
Effect of changing $k$ on spin procession.

Using the spin precession ansatz:

$$ \omega\delta S_i^y = JS(2\delta S_i^y \cos (\mathbf{k} \cdot \mathbf{a} ) - 2 \delta S_i^y ) $$

Spin wave dispersion:

$$ \omega = 2JS (\cos (\mathbf{k} \cdot \mathbf{a} ) - 1) $$